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Theorem eqglact 19210
Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x 𝑋 = (Base‘𝐺)
eqger.r = (𝐺 ~QG 𝑌)
eqglact.3 + = (+g𝐺)
Assertion
Ref Expression
eqglact ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Distinct variable groups:   𝑥, +   𝑥,   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
2 eqid 2735 . . . . . . 7 (invg𝐺) = (invg𝐺)
3 eqglact.3 . . . . . . 7 + = (+g𝐺)
4 eqger.r . . . . . . 7 = (𝐺 ~QG 𝑌)
51, 2, 3, 4eqgval 19208 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6 3anass 1094 . . . . . 6 ((𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
75, 6bitrdi 287 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
87baibd 539 . . . 4 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
983impa 1109 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
109abbidv 2806 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → {𝑥𝐴 𝑥} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11 dfec2 8747 . . 3 (𝐴𝑋 → [𝐴] = {𝑥𝐴 𝑥})
12113ad2ant3 1134 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = {𝑥𝐴 𝑥})
13 eqid 2735 . . . . . . . . 9 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
1413, 1, 3, 2grplactcnv 19074 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1514simprd 495 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
1613, 1grplactfval 19072 . . . . . . . . 9 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1716adantl 481 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1817cnveqd 5889 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
191, 2grpinvcl 19018 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
2013, 1grplactfval 19072 . . . . . . . 8 (((invg𝐺)‘𝐴) ∈ 𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2215, 18, 213eqtr3d 2783 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2322cnveqd 5889 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
24233adant2 1130 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2524imaeq1d 6079 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26 imacnvcnv 6228 . . 3 ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27 eqid 2735 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2827mptpreima 6260 . . . 4 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29 df-rab 3434 . . . 4 {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3028, 29eqtri 2763 . . 3 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3125, 26, 303eqtr3g 2798 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
3210, 12, 313eqtr4d 2785 1 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  wss 3963   class class class wbr 5148  cmpt 5231  ccnv 5688  cima 5692  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  invgcminusg 18965   ~QG cqg 19153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ec 8746  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-eqg 19156
This theorem is referenced by:  eqgen  19212  pzriprnglem10  21519  cldsubg  24135  tgpconncompeqg  24136  snclseqg  24140  ellcsrspsn  35626
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